FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES A Dynamic Model of Price Signaling, Consumer Learning, and Price Adjustment Matthew Osborne University of Toronto Adam Hale Shapiro Federal Reserve Bank of San Francisco November 2014 The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System. Working Paper 2014-27 http://www.frbsf.org/economic-research/publications/working-papers/wp2014-27.pdf
Transcript
FEDERAL RESERVE BANK OF SAN FRANCISCO
WORKING PAPER SERIES
A Dynamic Model of Price Signaling, Consumer Learning, and Price Adjustment
Matthew Osborne University of Toronto
Adam Hale Shapiro Federal Reserve Bank of San Francisco
November 2014
The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.
Working Paper 2014-27 http://www.frbsf.org/economic-research/publications/working-papers/wp2014-27.pdf
We develop a model of consumer learning and price signaling where price and quality
are optimally chosen by a monopolist. Price signaling may occur when consumers have
imperfect information about product quality. To infer quality, consumers may rely on
previous experience or may use some of the product’s observable characteristics, such as
the product’s price. We examine the scenario whereby the firm can endogenously change
consumers’ beliefs about the product’s quality by altering both the price and quality of
its product. Our main findings are that, in this type of setting, price signaling causes
the firm to raise its price, lower its quality, and dampen the degree to which it responds
to cost shocks. If the cost of adjusting quality is sufficiently high, the dampening effect
is pronounced in the downward direction, meaning that price signaling causes prices to
respond less to cost decreases than cost increases.
In our model, consumers have beliefs about the joint distribution of price and quality
for the firm’s product. Given no prior information on quality and price, consumer beliefs
about a firm’s quality may or may not be correct. Before making a purchase, consumers
do not observe the quality of the product but make an inference about it using their
beliefs. After purchase, the product’s quality is revealed and consumers update their
beliefs through a quasi-Bayesian learning process. Thus, over time consumers will observe
realized price and quality, and will update their beliefs about a firm’s policy given their
past experience. In each period, a forward-looking monopolist chooses the product’s
price and quality, accounting for the fact that its choice will affect consumer beliefs and
future profits. We solve the model using numerical methods and then simulate it under
a range of parameters.
To our knowledge, this study is the first to assess the effects of price signaling on price
adjustment with endogenous quality and a forward-looking monopolist.1 Previous studies
1Concurrent work by L’Huillier (2013) shows in a macroeconomic model with exogenous monetary
shocks and consumer learning that signaling can lead to price rigidity. In that model, the firm may have
an incentive to hide whether its product is cheap or expensive relative to the numeraire good by keeping
its price rigid. Two important differences between the two models are that in L’Huillier (2013) firms
are not forward-looking and the firm does not choose its quality. Our work accounts for the impact of
reputation on quality provision and price rigidity.
1
on price signaling, such as Judd and Riordan (1994) and Wolinsky (1983) have focused
on examining static equilibrium price levels and have assumed either a one- or two-period
setting. Our model extends this line of research by assessing the effects of cost changes on
price adjustment, but it also introduces a dynamic framework with consumer learning.
We show that consumer learning may be important if a firm’s optimal strategy includes
endogenously manipulating consumers’ expectations about the relationship between price
and quality. In particular, we find that this type of forward-looking behavior further
moderates the effect of cost changes on optimal prices. Due to the complexity of our
model, analytical solutions to the firm’s policy functions are not available. Thus, we
turn to simulation of a calibrated model to draw our conclusions.
Relative to a full information model, where a bit more than 50% of either a cost
increase or a cost decrease is passed on to prices, we find a signaling model implies that
only 43% of a cost increase and only 42% of a cost decrease is passed on. We solve our
model at parameter values that are calibrated to match aggregate margins by U.S. firms.
However, our finding that price signaling results in dampened pass-through is robust to
perturbations of these parameters. We also find evidence of asymmetric pass-through.
If the cost of adjusting quality is sufficiently high, a firm will pass on more of a cost
increase than a decrease, which is consistent with empirical findings (Peltzman 2000).
The results of our model have important insights regarding the literature on price
stickiness and cost pass-through, which covers a number of fields including marketing,
macroeconomics, and international economics. There are a number of explanations for
nominal price rigidity. Rotemberg (2005) proposes that price changes may be muted
because they may anger consumers. Cabral and Fishman (2012) put forth a model
showing that price changes may induce consumers to search for competitors’ products.
A few studies have shown that, more generally, prices will be more rigid the more curved
the demand function (see for example, Kimball (1995) and Klenow and Willis (2006)).
Indeed, in a recent study on cost pass-through in the coffee market, Nakamura and
Zerom (2010) find that this type of demand curvature is responsible for a large portion
of wholesale coffee price rigidity. Understanding what economic forces can drive price
rigidity is important from a policy perspective because price rigidities provide scope for
monetary policy. In macroeconomics, a large body of research is devoted to exploring
2
informational drivers of price stickiness (see Lucas (1972), Mankiw and Reis (2002), Reis
(2006), Mackowiak and Wiederholt (2009), and L’Huillier (2013)).
Building on prior work by Weyl and Fabinger (2013), we mathematically derive two
drivers of the impact of signaling on cost pass-through, which we denote as the static and
dynamic effects. The static mechanism is in line with studies by Kimball (1995), Klenow
and Willis (2006), and Nakamura and Zerom (2010). Specifically, we show that if the
steady-state ratio of quality to price is not too high, signaling increases the curvature of
the demand function, which ultimately acts to dampen how prices respond to changes in
cost. The demand function becomes more curved because, under price signaling, raising
the price may signal to the consumer that the product is of higher quality. Thus, demand
does not necessarily monotonically fall with prices. This finding may help explain the
relatively large degree of demand curvature needed to explain incomplete pass-through
by Nakamura and Zerom (2010).
The dynamic effect arises because a forward-looking firm recognizes that if it changes
the price of a product today, it affects consumer beliefs about the relationship between
price and quality in the future. Adjusting prices has two effects on future demand.
First, adjusting prices alters consumer beliefs about the relationship between price and
quality, changing the effectiveness of the signal. Second, adjusting prices changes the
precision of the signal, ”locking in” their beliefs in the future. If the firm’s value function
is sufficiently concave in price then signaling will lead to dampened pass-through. Our
simulation results suggest that the dampened pass-through rates from signaling are due
to both the static and dynamic effects. The static effect is much larger: signaling overall
dampens pass-through by about 8 or 9%, and about 7 to 8% of that is due to the
impact of signaling on the shape of the demand curve. To examine the generality of our
finding that signaling moderates pass-through, we derive conditions on the ratio of price
to quality under which signaling increases demand curvature. As long as this ratio is
above the consumer’s willingness to pay for the product, and below an upper bound that
increases in the share of income spent on the product, signaling will increase the demand
curvature. Our findings suggest that the dampening effect of signaling on pass-through
is more likely to be prevalent in more expensive products.
It is important to note that our modeling framework differs from prior theoretical
3
work on signaling such as Wolinsky (1983). This literature has assumed that consumers
know the functional form of demand and costs, and can therefore directly compute the
firm’s optimal pricing strategy. While this approach is tractable in the static settings
typically examined in the signaling literature, it is intractable in a dynamic setting such
as ours. Therefore, a strong assumption of our model is that consumers do not base
their beliefs on the optimal solution of the firm’s problem. Similar issues have arisen in
the estimation of dynamic games.2 In our context, consumers are following a behavioral
rule that is consistent with the notion of equilibrium behavior proposed by Fershtman
and Pakes (2012). A consequence of our assumption is that, in our model, consumers
will never be able to learn the optimal policy of the firm exactly as they would in more
standard models. However, due to the Bayesian updating process they will, on average,
be able to correctly predict the firm’s quality conditional on price.
In addition to contributing to the literature on price stickiness, our paper also con-
tributes to the literature that models a firm’s reputation for quality. Klein and Leffler
(1981) and Shapiro (1983) assess the conditions under which firms do not “cheat” con-
sumers by offering a quality that is less than contracted for. Both studies consider a
forward-looking firm and find that prices are set at a premium above cost in order to
compensate for the firm’s initial investment in quality.3 Similar to our study, these stud-
ies consider forward-looking firms; however, they differ from ours in that they do not
explicitly model the beliefs of consumers. These authors assume that once a firm cheats
(by offering a low quality product at a high price) all consumers cease to purchase from
the firm—a type of trigger strategy. More recent studies on reputation such as Board
and Meyer-ter Vehn (2013) model quality as a function of past investments in quality,
implicitly allowing the firm to control consumers’ beliefs through persistent investment
decisions. In our model, a firm has an incentive to cheat consumers because they do not
immediately observe a product’s quality and must learn about it over time. Consumers
in our model learn according to a modified Bayesian learning process where consumer
forgetting can occur: observations that occurred far in the past may be downweighted
2Recent work, for example Fershtman and Pakes (2012), has proposed that, rather than being fully
rational, some agents are assumed to use past outcomes to forecast current outcomes.3Specifically, a large discounted future flow of profits can then be a better strategy than pocketing
the short-term gain of reducing quality and losing future customers.
4
relative to more recent data (see, for example, Orphanides and Williams (2004)).4 We
find that a firm has more incentive to cheat consumers if they forget the past more eas-
ily; if the firm produces a poor quality product beliefs will update quickly. In contrast,
if consumer beliefs update slowly then there is scope for a firm to offer a high quality
product for a while to raise consumer perceptions, and then to cheat consumers for some
time by producing a low quality product at a high price.
An outline of the paper is as follows. In Section 2, we introduce our model of price
signaling and consumer learning. Section3 describes the techniques used to numerically
solve the model, and the steady state of our model at our preferred parameterization.
Section 4 shows the simulated pass-through from our preferred parameterization, the
mathematical derivations of pass-through, and the breakdown of pass-through into static
and dynamic components. Section 6 shows the robustness of our results to alternative
parameterizations, and Section 7 concludes.
2 Model
2.1 Demand
We assume there exists a continuous measure of infinitely lived consumers indexed by
i, each endowed with wealth wi ∼ U [0, w]. The total measure of consumers is normalized
to 1. Under an additively linear direct utility function, it follows that consumer i’s
indirect utility upon purchase of a product with quality xt and price pt in time period t
is:
Ui = xt − αpt + ǫIt. (1)
The error term ǫit is an i.i.d. taste shock that reflects consumer heterogeneity in tastes
for the firm’s product. ǫit is observed to the consumer at time t but unobserved to the
firm. We assume that the ǫit follows a type 1 Extreme value distribution. α measures the
consumer’s marginal utility of income.5 If a consumer does not purchase the product,
4In standard Bayesian updating, all observations receive equal weight.5In this model we implicitly assume that the consumer has a utility function that is quasilinear in two
goods, the monopolist’s product and a numeraire good z. If the consumer purchases the monopolist’s
5
she consumes the outside good. We normalize the price and quality of the outside good
to zero.
The case we explore in this study consists of the scenario where the consumer does
not know the true quality of the product, xt, and thus must infer the quality based on
her information set ℑt (that is, her beliefs).6 Consumer i purchases the product if her
expected utility for the product is positive, and if the price is lower than her wealth wi.
Under these assumptions, the aggregate demand for the good will be
D(pt,ℑt) =
{ (1− pt
w
) (exp(E(xt|ℑt)−αpt)
1+exp(E(xt|ℑt)−αpt)
)
if pt ≤ w
0 otherwise.(2)
We assume that the consumers believe the expected quality conditional on the observed
price is linear in the price:
E(xit|pit) = b0 + b1p. (3)
Because in our model we interpret the outside good as a product with quality 0 and
price 0, in order for consumer expectations to be consistent we normalize b0 = 0. We
additionally assume that consumers do not know b1, but learn about it over time. At
the beginning of period t, consumers observe only the price and condition their beliefs
about product quality on price and on their beliefs about b1. We denote the consumer’s
expected value of b1 as β, and the variance of their belief around β as σ2. At the end
of the period, the quality of the firm’s product is revealed to the entire market. The
information revelation could occur through magazine ratings such as Consumer Reports,
ratings on Internet websites such as Amazon, or word of mouth.7 We assume that
consumers form their beliefs in period t using a weighted regression:
βt = minβ
t∑
τ=1
λ(t−τ)(xτ − βpτ )2. (4)
good then her consumption of the numeraire good is z = wi − p.6For reasons we will clarify below, in our model the information set will be the same for all consumers.7Allowing for consumers to differ in their beliefs would complicate the modeling considerably. The
firm would have to track an infinite dimensional state variable, the distribution of beliefs, over time.
6
The term λ ≤ 1 is an exponential forgetting factor that downweights earlier obser-
vations. If λ = 1, then consumers have perfect recall; λ < 1 assumes that consumers
eventually forget early observations. Note that if λ = 1, then consumer uncertainty
about the value of b1 will approach zero. This can lead to the somewhat unrealistic
scenario where, if a firm were to choose a high quality and a high price for a long pe-
riod of time, consumer beliefs would not update much if the firm suddenly dropped its
quality to zero. If λ < 1, consumers will always remain somewhat uncertain about the
relationship between price and quality and a large change in x will have an impact on
consumer beliefs, even in the limit (t = ∞).
When we solve the problem, it is convenient to write consumer belief updating in re-
cursive form as a Kalman filter following Bersetkas and Tsitsiklas (1996). If the variance
of beliefs at time t is σ2t−1 then the updating formulas for βt and σ2
t can be written as:
σ2t =
1λ
σ2t−1
+ p2t(5)
βt = βt−1 + σ2t pt(xt − βt−1pt).
Note that if λ = 1, it can be shown that s2 → 0 as t → ∞.
The effect of changing prices on consumer beliefs leads to the dynamic effects of
signaling which we discuss in the introduction. Consumer beliefs will update so that, on
average, they are correct. For a given quality level of the firm, a price increase will tend to
lower the parameter βt, weakening consumer beliefs about the relationship between price
and quality. Additionally, a higher price provides a stronger signal to consumers, and
will lead to a lower value of σ2t−1, as shown in equation (6). To the firm, the advantage
of having a lower value of σ2t−1 is that consumers will be more locked in to their beliefs in
the future, and their beliefs will update more slowly to future price and quality changes.
Before proceeding we make some comments on our assumptions about consumer
beliefs and the process by which beliefs update. First, we note that we assume consumers
believe the expected value of quality is linear in price. This assumption is attractive
due to its tractability and has been used in learning papers in macroeconomics (Wieland
2000). One way to interpret this assumption is that, although the actual function E(x|p)
may be nonlinear in price, consumers are boundedly rational and have a limited ability
7
to process information. One could approximate E(x|p) with any degree of accuracy
using a Taylor expansion; the higher is the order of this Taylor expansion, the more
parameters one has to track. Our assumption would be justified by the assumption that
consumers use a linear approximation. To check the robustness of our findings to this
assumption, in Appendix 11 we consider an alternative model where prices and quality
take on discrete values, and consumer beliefs are not constrained to be linear. Our overall
findings appear to be robust to the specification of beliefs. We prefer a specification of
beliefs that allows for continuous prices, because observed prices are continuous and
pass-through derivations are more tractable in a continuous setting.
A second assumption underlying consumer behavior is that consumers do not solve the
firm’s dynamic problem and base their beliefs on the optimal solution of the firm problem.
Prior theoretical work on signaling such as Wolinsky (1983), has assumed that consumers
know the functional form of demand and costs, and can compute a firm’s optimal pricing
strategy. Consumers make purchase decisions knowing this optimal strategy and taking it
into account. This approach is tractable in the static settings that are typically examined
in the signaling literature, but would not be tractable in a setting such as ours. Similar
issues have arisen in the estimation of dynamic games, and recent work in that area has
proposed that rather than modeling all agents as being fully rational, one can assume
that some agents use outcomes experienced in past periods to form forecasts of outcomes
in current periods (Fershtman and Pakes 2012). In our context, consumers are following
a behavioral rule that is consistent with the notion of equilibrium behavior proposed by
Fershtman and Pakes (2012).
2.2 Firm
A monopolist maximizes profits in a dynamic sense, in that it chooses price and
quality, keeping in mind how this joint decision alters consumers’ expectation of quality.
We assume that the firm’s marginal cost, ct, is a stochastic vector (c0t, c1t). Each draw of
ct is i.i.d. across time with distribution Fc(·). The stochastic nature of cost can include
a multitude of exogenous factors such as process innovations, weather disruptions, or
factory malfunctions. Firm profits will be a function of price, quality, cost draws, and
consumer beliefs at time t, which we denote as St = (βt, σ2t ):
We assume that the policy step has converged when maxi=1,Nj
s‖V p
np+1(sji )−V p
np(sji )‖ <
ǫp, where we choose ǫp = 1e−4. The policy iteration step in equation (31) converges very
quickly. Once the policy step converges, we set Vn = V p, and solve for the optimal policies
at step n+ 1 using equation (30). The algorithm converges when maxi=1,Ns‖Vn+1(si)−
Vn(si)‖ < ǫv, where we set ǫv = 1e − 3. We have found that this algorithm converges
much more quickly than standard value function iteration, where one would solve for the
optimal policy every time one updated the value function.
3.1 Model Simulation
We simulate equilibrium prices and qualities for two versions of this model, one with
zero adjustment cost and one with infinite adjustment cost. These two cases allow us to
infer how price signaling affects the firm’s pricing decisions in different types of situations.
An infinite adjustment cost model might approximate the case in which a firm initially
decides whether it wants to sell in the luxury or the economy market. A zero adjustment
cost model, on the other hand, might represent a barber who can adjust quality by simply
exerting different amounts of effort. The case with an infinite adjustment cost is simpler
11
in that the firm never adjusts the quality of its product, only its price. Therefore, once
quality is initially chosen it can be considered exogenous thereafter. In the case, of zero
adjustment costs, the firm is free to adjust both its price and quality without any cost.
That is, quality is considered endogenous.
Our preferred parameter values for numerical solution of the model are shown in
Table 1. Maximum income is normalized to 1, and the firm’s discount factor is chosen
to be consistent with a yearly interest rate of 5%. The cost parameter values are chosen
to produce prices such that marginal costs match aggregate labor share of income in the
U.S. economy, which is 60% (this approach to calibrating markups is often used in papers
in macroeconomics, for example Gali and Gertler (1999)). In this sense our parameter
values are realistic in that they imply average markups that are consistent with what
is observed for an average firm. The parameter on the convexity of marginal costs in
quality is chosen to be 2. To get a sense of the robustness of our conclusions to changes
in the parameters, we will also solve the model for cost distributions that are lower and
higher than those in the table, and for values of λ that are lower and higher than 0.99.
We discuss these perturbations to the model in Section 6.
The solution to the firm’s problem involves computing an optimal price p and an
optimal quality x. To simplify the solution of this problem we discretize the x values
and assume that the firm can choose x in increments of 0.1 between 0.5 and 10. The
distribution of marginal costs is assumed to be uniform, and we integrate over costs using
Gauss-Legendre quadrature with five quadrature points.
3.1.1 Fixed Quality: Infinite Adjustment Costs
The policy function under the infinite adjustment cost case is depicted in Figure 1.
We graph the policy function conditional on the quality x = 3.6; as we show below, this
is the level of quality that maximizes profits if consumers have full information and know
the quality of the product with certainty. The figure shows that at very low values of β,
price is increasing in β. At higher values of β, the optimal price becomes relatively flat
with respect to β.
The non-monotonicity of the pricing policy function in β seems counterintuitive but
is attributable to the fact that increasing β has two effects on demand. First, for a given
12
Table 1: Parameter Values for Base Simulation
Variable Value
δ 0.95
w 1
α 1
c0 U[0.35, 0.45]
c1 0.002
ν 2
λ 0.99
price, a larger β signals that the product is of higher quality, so demand is higher. All
else equal, this “demand effect” should induce the firm to raise its price. Second, a larger
β means that a change in price will have a larger effect on the change in demand, which
acts to increase the price elasticity of demand. All else equal, this “elasticity effect”
will induce the firm to lower its price. When β is low, an increase in β has a relatively
larger demand effect than an elasticity effect, so the price increases in β. However, at
higher levels of β the elasticity effect dominates slightly, which is why the pricing policy
function eventually starts decreasing in β. In Appendix 9, Figure 3, we show how the
optimal price for a myopic firm is affected by changes in β.
We assume the firm chooses a level of quality in period 0 to maximize the expected
present discounted value of profits, and cannot change it afterwards. The optimal level
of x depends on consumer beliefs. In the steady state of this model, beliefs and prices
will always adjust to the same values conditional on x, so the effect of the starting points
will die out. We simulated firm decisions at starting values of β = 3.5 and σ2 = 1, which
produced an optimal x of 3.6, which is the same as the optimal x in the full information
(i.e. a model with no signaling) case.9
9In our simulations we have found that the optimal level of x is decreasing in the initial level of
β, and is also decreasing in σ2. The higher is β and the lower is σ2, the lower will be the optimal x.
Producing a higher quality product is costly for the firm in the short run; however, the benefits to being
13
0
5
10
15
20
5
10
15
0.4
0.5
0.6
0.7
0.8
0.9
1.0
βσ2
p
Figure 1: Pricing policy function for xt−1 = 3.6, γ = ∞
Table 2 shows the steady-state values of price, quality, firm profit, and consumer
surplus.10 To assess how price signaling impacts price and quality over and above the
standard textbook case, the table also shows the case where the consumer knows the
quality of the product with certainty (i.e., the case of full information). The general
takeaway from the table is that information asymmetry (i.e., price signaling) acts to
raise the equilibrium price relative to full information and also lowers consumer surplus.
3.1.2 Flexible Quality: Zero Adjustment Costs
The policy functions under the zero adjustment cost case are depicted in Figure 2.
Here, there are two policy functions (one for x and one for p) since the choice of x is
high quality are only realized in the long run as consumers adjust their beliefs about the relationship
between price and quality. If β is initially higher, the firm has more incentive to produce a lower quality
product since it will take longer for consumers to figure out the firm is bad; similar results arise if σ2 is
lower.10The values of outcome variables are the averages over the simulation, starting with the 500th
iteration and ending in the 1000th. The first 500 iterations are removed for burn-in. For a graphical
illustration of the steady state values see Appendix 10.
14
endogenous. The shape of the pricing policy looks similar to the γ = ∞ case, with some
some subtle differences. This has to do with how the choice of x impacts the choice
of p. First, note that the optimal choice of quality is generally decreasing in β. The
intuition here is that as β becomes smaller, the consumer is less perceptive that a higher
price signals higher quality. Thus, when β decreases, the firm attempts to raise the price
signal by choosing to offer a high quality product at a high price. As β becomes larger
this effect is weaker, hence the optimal price now falls with β when β is large. It is
interesting to note how the optimal choice of x is impacted by σ2. The optimal choice
of x is generally increasing in σ2. The intuition here is that, the smaller is σ2, the more
apt the firm is to exploit the signal by cheating. For instance, if the consumer is very
confident that a high price signals a high quality, areas where σ2 is very low, the firm
will cheat the consumer by offering a very low quality product at a relatively high price.
0
5
10
15
20
5
10
15
0.4
0.5
0.6
0.7
0.8
0.9
1.0
βσ2
p
0
5
10
15
20
5
10
15
2
4
6
8
10
βσ2
x
Figure 2: Policy functions for γ = 0
As shown in Table 2, the effect of signaling on price is exaggerated in the case of
flexible quality. Specifically, the firm chooses to offer a lower quality product, x = 2.3,
at the similarly higher price found in the fixed quality setting.11 We should note that,
11We simulate the firm’s policy and start out at state variables of β = 3.2 and s = 0.019, which are
15
although average steady-state profits fall a bit moving from γ = 0 to γ = ∞, the present
discounted value of future profits (not shown in the table) rises.12 These results are
robust to different parameters, as we discuss in Section 6.
Table 2: Comparison of Full Information and signaling Steady States
γ = ∞ γ = 0
Full Info signaling Full Info signaling
Price 0.71 0.72 0.71 0.72
Quality 3.6 3.6 3.64 2.3
Firm Profit 0.08 0.08 0.08 0.07
Consumer Surplus 0.85 0.82 0.86 0.5
In both full information cases, all the outcome variables (profits, prices,
etc.) are averaged over costs. The values of outcome variables for the
signaling model are the averages over the simulation, starting with the
500th iteration and ending with the 1000th. The first 500 iterations are
removed for burn-in.
4 The Impact of Price Signaling on Cost Pass-through
We assess how price signaling affects price adjustment by simulating the model under
a temporary cost shock. We begin at a base cost of 0.40, and for a decrease we drop
the cost to 0.35, while for an increase we raise the cost to 0.45. We solve for optimal
prices both before and after the cost change. Simulating both a cost increase and a cost
decrease allows us to examine whether price signaling has any impact on the symmetry
of pass-through.
the long-run values when x is fixed at 3.6. The steady state of the γ = 0 specification does not appear
to be affected by the starting points of the simulation — we have always gotten to the same steady state
for different initial beliefs.12When the firm can choose quality, it initially chooses a slightly lower quality product to exploit
beliefs. It subsequently raises quality over time to 2.3 as beliefs update (i.e., as β falls). In the γ = ∞
case, profits seem to be relatively constant over time.
16
The results of this exercise are depicted in Table 3. Specifically, this table shows
the percentage impact of a cost shock on prices. We depict the case for x = 2.3—
the equilibrium quality under the case of zero adjustment costs—as well as the case
for x = 3.6—the equilibrium quality under the case of infinite adjustment costs. For
completeness, we also show the infinite adjustment cost case under the assumption that
x = 2.3. The table also shows cost pass-through when we shut down the dynamic
component of the model (i.e., when we set δ = 0), which we label the myopic model.
Relative to the full-information case, price signaling acts to considerably decrease
cost pass-through. This effect is more pronounced for a cost decrease relative to a cost
increase. For instance, in the case of x = 3.6, price signaling causes pass-through to fall
from 51% to 46% for a cost increase, but to 45% for a cost increase. As we will discuss
later, most of the overall effect from signaling is coming from the myopic component of
the model—the impact of signaling on the shape of the demand curve.
The asymmetry of cost pass-through seems to reverse when quality is chosen endoge-
nously (i.e., γ = 0.) In this case, pass-through falls from 52% to 44% for a cost increase,
but to 45 percent for a cost decrease. This reversal of the asymmetry likely arises due
to the fact that the firm also adjusts quality in response to cost shocks. When cost
decreases, the firm drops quality from 2.3 to 2.1. However, when cost increases, quality
rises, but by a smaller amount to 2.4. Prices will tend to move with quality, and since
quality drops more in response to cost decreases, prices do as well. Another interesting
result is that pass-through seems to be lower for higher quality goods.
5 Discussion of Pass-Through Results
Our results indicate that price-signaling generates a considerable amount of price
rigidity. In the following subsection we will discuss the role of demand-curvature and
how this mechanism explains the link between price signaling and cost pass-through.
We explain the intuition analytically under the simpler case of fixed quality, and then
show that the intuition holds in the case where the firm can choose quality (i.e., flexible
quality).
17
Table 3: The Impact of a Cost Shock on Prices
x = 2.3 x = 3.6
Full Info Signaling Full Info Signaling
Dynamic Myopic Dynamic Myopic
γ = ∞ Cost Decrease 52 42 43 51 45 45
Cost Increase 52 43 44 51 46 46
γ = 0 Cost Decrease 52 45 51
Cost Increase 52 44 51
Notes: This table shows the percentage impact of a cost increase or decrease on prices. For all the
simulations, we begin at a base cost of 0.40, and for a decrease we drop the cost to 0.35. For an
increase we raise the cost to 0.45. In the signaling cases, we simulate the model for 1000 periods,
holding cost fixed at its new value. That is, in period 1000 the cost is 0.40, and in the following
period cost is 0.35 or 0.45.
5.1 Fixed Quality: Infinite Adjustment Costs
The simplest case considers a high adjustment cost (γ) so that the firm chooses quality
in period 1 and never changes it after that. We will derive the impact of an idiosyncratic
shock to c0 on price. Recall that we assume that the stochastic process for the evolution
of costs is
cjt ∼ U [cj , cj ],
where cjt is i.i.d. over time and j. To explain the intuition behind how cost shocks affect
prices, we first assume that the firm is myopic δ = 0, that γ is high so x is fixed over
time, and that costs are nonstochastic: cjt = cj = cj, ∀j, t. We can derive the impact
of an increase in c0 on prices following analysis outlined in Weyl and Fabinger (2013)
and Jaffe and Weyl (2013). We write the firm first-order conditions as
f(p) = −∂D
∂p
−1
D(p)− (p− c0 − c1xα) = 0. (11)
If we add a tax t to the marginal cost, the first-order condition becomes
18
f(p) + t = 0.
By implicitly differentiating this first-order condition we get the equation for static pass-
through:
∂p
∂t=
−1
f ′(p). (12)
Note that f ′(p) = D ∂2D∂p2
/(
∂D∂p
)2
− 2 depends only on the first and second derivatives
of demand, and is a measure of the curvature of the demand curve. This provides one way
for signaling to affect pass-through. For the case of full information, where x is known
to consumers, we could compute pass-through from equation (12). If signaling increases
the curvature of demand relative to full information, then that increase in curvature will
dampen pass-through. Intuitively, we would expect signaling to dampen pass-through in
the short term, especially for cost decreases. If the monopolist observes a cost decrease,
it has a disincentive to drop the price because a lower price signals lower quality.
The dynamics in our model provide an additional avenue for pass-through. Suppose
that the firm is forward-looking, so δ > 0. Then we can write the firm’s first-order
conditions in the same way as equation (11):
−∂D
∂p
−1
D(p)− (p− c0 − c1xα)
︸ ︷︷ ︸
f(p)
−δ∂D
∂p
−1∂V
∂S
′∂L
∂p︸ ︷︷ ︸
g(p)
+t = 013
An addition term g(p) arises which is the derivative of the value function with respect
to price.14 Note that this value function derivative is a function of t directly, since if we
change the tax on costs, we change firm future profits. The dynamic analogue to our
static pass-through equation (12) is
∂p
∂t= −
1
f ′(p) + ∂g
∂p
. (13)
13The second term, g, is not a function of t because we are imagining a temporary cost shock. If the
shock to costs were permanent this term would be a function of t as well.14The partial of V is the Jacobian of V with respect to its arguments, and the L partial is a Jacobian
as well. The prime indicates transpose, not a future value.
19
The term ∂g
∂pcaptures the dynamic effect of signaling on pass-through; if this term is
negative, then signaling further dampens pass-through (in full information the firm is
not forward-looking, so ∂g
∂p= 0). The sign of this derivative will depend on the signs
of the first and second derivatives of the value function with respect to price, and on
the curvature of demand. To see how, note that if we express the value function as a
function of price (subsuming the state transition) we can write
∂g
∂p=
∂2D∂p2
(∂D∂p
)2
∂V ′
∂p−
1∂D∂p
∂2V ′
∂p2. (14)
Note that the first term in equation (5.1) contains the demand curvature from the
static pass-through equation. We have previously argued that this term will be negative
under signaling. Furthermore, for a sufficiently high price ∂D∂p
will be negative as well un-
der signaling.15 For equation (5.1) to be negative under pass-through, the value function
must be increasing and concave in price.
Table 4: Analytical Derivatives that Determine Pass-Through (γ = ∞)
x=2.3 x=3.6
Full Info Signaling Full Info Signaling
f ′(p) -1.93 -2.18 -1.98 -2.11∂g
∂p-0.028 -0.010
We found in Table 3 in the previous section that when x was fixed, prices were more
rigid for a forward-looking firm than for a myopic one. We can relate this finding to
the analytical equations (12) and (5.1) by computing the numerical equivalent in our
simulation. These are shown in Table 4. Because f ′(p) falls from -1.93 to -2.18, signaling
increases the curvature of demand acting to dampen pass-through. Additionally, because
15It is possible for ∂D∂p
to be positive under signaling for very low prices, because higher prices imply
higher quality, increasing demand. The inclusion of the term 1 − pw
in our demand equation ensures
that even if signaling is very strong, demand will eventually be decreasing in price.
20
∂g
∂p< 0 the firm’s forward-looking behavior acts to further dampen pass-through. The
magnitude of the static effect is much larger than the dynamic effect.
5.2 Flexible Quality: Zero Adjustment Cost
We now consider the case where x may also respond to changes in cost. In our
simulations we discretized x, so if the cost shock is sufficiently small the firm will not
change x, and the analysis should be similar to the previous section. However, for a large
enough cost shock, x will change. When thinking about the change in x, the interesting
case is where the firm is forward-looking—if the firm is myopic, it will set x as low as
possible because increasing x does not affect current demand, but does increase marginal
cost. The math is similar to the previous case, although there are some complications.
For this exercise, we will assume that the set of x’s the firm can choose is continuous:
x ∈ [0,∞].
When the firm can adjust x, there are two first-order conditions. The first one is the
same as before, but we will relabel f as f1 and g as g1. The second equation will have
two similar terms,
f2(p, x) = −D(p)αc1xα−1
g2(p, x) = δ∂V
∂S
′∂L
∂x,
and we can write the system of equations that defines the pass-through rate as
f1(p) + g1(p, x) + t = 0
f2(p, x) + g2(p, x) = 0.
It is straightforward to show that two pass-through rates arise from this system, one for
p and one for x. They are
21
∂p
∂t=
−1
f ′1 + g1p − g2pg1x/(f2x + g2x)
(15)
∂x
∂t=
f2p + g2p(f ′
1 + g1p)(f2x + g2x)− g2pg1x. (16)
The price pass-through equation is similar to the infinite adjustment cost case, except
that the denominator contains an extra term that involves the derivatives of the first
order condition with respect to x. Note that if the firm is myopic, the g terms will be zero
in equation (15), and the equation will collapse to equation (5.1). Similarly, equation
(16) collapses to −∂p
∂t
f2pf2x
which is negative.16
The logic of the previous paragraph suggests that one can bread down the impact of
pass-through into static and dynamic components as we did in Table 4. Under signaling,
f ′1(p) = −2.17, and the static pass-through rate under signaling will be 1/2.17 = 0.46.
Since the overall pass-through rate under signaling is between 0.44 and 0.45 (Table 3), we
can conclude that the optimal policy of a forward-looking firm is to dampen pass-through
more than a myopic firm would.17
5.3 Generality of the Impact of Signaling on Pass-through
The results in Table 3, and our derivations of f ′(p) in the previous section, show that
the most important impact of signaling on pass-through occurs through shape of the
demand curve, measured by the term f ′(p). A follow-up question we now consider is:
under what general conditions will signaling reduce the term f ′(p)? Note that in Table
3, f ′FullInfo(p) is somewhat larger than -2 and f ′
Signaling(p) is lower than -2. Hence, two
sufficient conditions for signaling to result in dampened pass-through are
16Since ∂p∂t
> 0 and f2p < 0 and f2x < 0, we should get ∂x∂t
< 0, which is intuitive.17Although, as with γ = ∞, the bulk of the impact of signaling on pass-through operates through the
static effect.
22
f ′FullInfo(p) + 2 = D
∂2D
∂p2/
(∂D
∂p
)2
> 0
f ′Signaling(p) + 2 = D
∂2D
∂p2/
(∂D
∂p
)2
< 0.
It can be shown that in the full information model the static pass-through term f ′(p)
reduces to
f ′FullInfo(p) + 2 = wα
(1− p
w
)1
1+exp(x−αp)
(
2 + wα(1− p
w
) 1−exp(x−αp)1+exp(x−αp)
)
(
1 + wα(1− p
w
)1
1+exp(x−αp)
)2 , (17)
and in the signaling model this is
f ′Signaling(p) + 2 = −
(β − α)(1− p
w
)1
1+exp((β−α)p)
(2w− (β − α)
(1− p
w
) 1−exp((β−α)p)1+exp((β−α)p)
)
(1w− (β − α)
(1− p
w
)1
1+exp((β−α)p)
)2 .
(18)
First consider equation (17). This equation will be positive as long as
2 + wα(
1−p
w
) 1− exp(x− αp)
1 + exp(x− αp)> 0.
At our chosen normalization of α = 1 and w = 1, this inequality will always hold because1−exp(x−αp)1+exp(x−αp)
> −1. In general, as long as the term α that measures the marginal utility
of income is not too large relative to the maximum wealth, the inequality will hold.
Now let us turn to equation ((18)). A sufficient condition for this to be negative is if
(β − α)
(2
w− (β − α)
(
1−p
w
) 1− exp((β − α)p)
1 + exp((β − α)p)
)
> 0.
This result suggests that signaling is more likely to dampen pass-through for relatively
expensive goods—goods that take up a significant share of income. To see this, note that
this inequality will hold as long as β is in the range [α, 2/(w−p)+α]. In the steady state
23
of the signaling model, β will be close to x/p, the ratio of quality to price. What this
inequality tells us is that the quality to price ratio should be larger than the consumer’s
willingness to pay for quality, α, but not too much larger as it depends on the inverse
of the consumer’s remaining budget—w − p. That is, the upper bound on β will rise as
the share of income spent on the monopolist’s product increases.18
6 Pass-through at Alternative Parameters
In this section, we investigate the robustness of our results by altering the parameters
of the model. Specifically, we assess the model under alternate distributions of costs and
the exponential forgetting factor λ. In Table 5 we show the full information and signaling
steady states for four different parameterizations of the model: a version with a lower
cost distribution of U [0.2, 0.3], a higher cost distribution of U [0.5, 0.6], a low value of the
exponential forgetting factor, λ = 0.95, and a high value of λ = 0.999.
Overall, Table 5 shows an emerging pattern similar to that found in our previous
analysis. Relative to the case of full information, signaling causes consumer surplus to
fall—the firm lowers the quality of its product and sells it at a higher or similar price.
This is especially true in the case where the firm can freely adjust quality. An exception
is the case where the cost distribution is high and quality is fixed (i.e., the case of infinite
adjustment costs). In this instance, the signaling and full-information cases result in
similar steady states.
Table 6 shows how pass-through responds to changes in the cost distribution and
λ. There are three findings that are robust to changes in the cost distribution and λ:
First, signaling results in dampened pass-through relative to full information. Second,
18At our chosen parameter values, note that the bounds are approximately [1, 7.7], and the quality
to price ratio is around 3 to 5 (Table 2). The inequality that defines how signaling affects pass-through
looks like it depends on the units used to measure wealth due to the inclusion of the term 2/(w − p).
This is not the case however, for two reasons. First, if we scaled up wealth and prices by some factor
κ, we would have to scale down the marginal utility of income, α, by the same amount. Additionally,
since the term β measures how many units of utility a consumer expects to get from a product of
price p, β would also scale by the same factor in the steady state. The inequality would then become
β/κ ∈ [α/κ, 2/(κw − κp) + α/κ].
24
Table 5: Robustness: Comparison of Full Information and Signaling Steady States
γ = ∞ γ = 0
Full Info Signaling Full Info Signaling
c0t ∼ U [0.2, 0.3]
Price 0.64 0.65 0.64 0.65
Quality 3.8 3.6 3.76 2.35
Firm Profit 0.12 0.12 0.12 0.12
Consumer Surplus 1.16 1.04 1.15 0.65
c0t ∼ U [0.5, 0.6]
Price 0.79 0.79 0.79 0.79
Quality 3.5 3.6 3.44 2.03
Firm Profit 0.04 0.04 0.04 0.04
Consumer Surplus 0.6 0.6 0.58 0.32
λ = 0.95
Price 0.71 0.72 0.71 0.71
Quality 3.6 3.6 3.64 3.14
Firm Profit 0.08 0.08 0.08 0.08
Consumer Surplus 0.85 0.82 0.86 0.72
λ = 0.999
Price 0.71 0.72 0.71 0.71
Quality 3.6 3.6 3.64 0.95
Firm Profit 0.08 0.08 0.08 0.06
Consumer Surplus 0.85 0.82 0.86 0.23
In both full information cases, all the outcome variables (profits, prices,
etc.) are averaged over costs. The values of outcome variables for the
signaling model are the averages over the simulation, starting with the
500th iteration and ending with the 1000th. The first 500 iterations are
removed for burn-in.
25
pass-through rates are slightly higher when the firm is myopic. Third, pass-through is
asymmetric when the cost of changing quality is high. In contrast, with low adjustment
costs, the asymmetry of pass-through varies with λ and with the cost distribution.
Recall that we found in the last section that when adjustment costs are low, the firm
passes on more of cost decreases than increases. We can see in the second column of
Table 6 that for low and high cost distributions the firm passes on more of cost increases
than decreases. When λ is low, the firm passes on more of cost decreases than increases;
this finding reverses when λ is 0.999. Overall, it appears that the asymmetry of pass-
through depends on both the cost distribution and the speed at which consumer beliefs
update. For extreme values of these parameters firms pass on more of cost increases than
decreases. For values that are more in the middle we observe the opposite.
7 Conclusion
Our results indicate that price signaling affects the optimal price, quality, and the
rate at which the optimal price adjusts after a cost shock. Our result that price signaling
acts to raise the equilibrium price relative to the case of full information, corresponds
well with classical works on price signaling, for example, Wolinsky (1983). Our model,
however, extends this line of literature by allowing the firm to choose quality, and there-
fore dynamically affect the beliefs of the consumer about the price signal. We find that
under this type of setting the firm chooses to offer a lower quality product at higher
price relative to the full information case. We contribute to the literature examining the
drivers of price stickiness by deriving conditions under which signaling leads to dampened
pass-through.
There are many ways in which our work could be extended. One possibility would
be to examine how results might change under competition. If more competition tends
to result in decreased prices and increased quality, then the impact of signaling on pass-
through could be mitigated. It could also be interesting to examine the robustness of
our conclusions to the functional form used for the demand curve. We think that our
proposed functional form is a reasonable starting point because it is based on the logit
demand model, which is standard in empirical work. Finally, another step would be to
26
Table 6: Robustness: The Impact of a Cost Shock on Prices
x = 2.3 x = 3.6
Full Info signaling Full Info signaling
Dynamic Myopic Dynamic Myopic
c0t ∼ U [0.2, 0.3]
γ = ∞ Cost Decrease 52 39 40 51 42 43
Cost Increase 52 40 41 51 43 44
γ = 0 Cost Decrease 52 36 51
Cost Increase 52 45 51
c0t ∼ U [0.5, 0.6]
γ = ∞ Cost Decrease 52 45 45 51 47 47
Cost Increase 52 46 46 51 47 47
γ = 0 Cost Decrease 52 47 51
Cost Increase 52 49 51
λ = 0.95
γ = ∞ Cost Decrease 52 42 43 51 44 45
Cost Increase 52 43 44 51 45 46
γ = 0 Cost Decrease 52 50 51
Cost Increase 52 40 51
λ = 0.999
γ = ∞ Cost Decrease 52 43 43 51 45 45
Cost Increase 52 44 44 51 46 46
γ = 0 Cost Decrease 52 45 51
Cost Increase 52 46 51
Notes: This table shows the percentage impact of a cost increase or decrease on prices. For all the
simulations, we begin at a base cost at the midpoint of the c0t distribution, and for a decrease we
drop cost to the distribution’s lower bound. For an increase we raise cost to its upper bound. In the
signaling cases, we simulate the model for 1000 periods, holding cost fixed at its new value. That
is, in period 1000 the cost is 0.5 ∗ (c+ c) and in the following period cost is c or c.
27
take our model to the data and to determine the prevalence of signaling in actual markets.
This question has received attention in the marketing literature (Erdem, Keane, and Sun
2000).
28
References
Bersetkas, D. and J. Tsitsiklas (1996). Neuro-Dynamic Programming. Belmont, MA:
Athena Scientific.
Board, S. and M. Meyer-ter Vehn (2013). Reputation for quality. Econometrica 81 (6),
2381–2462.
Cabral, L. and A. Fishman (2012). Business as usual: A consumer search theory of
sticky prices and asymmetric price adjustment. International Journal of Industrial
Organization 30 (4), 371–376.
Erdem, T., M. Keane, and B. Sun (2000). A dynamic model of brand choice when
price and advertising signal product quality. Marketing Science 27 (6), 1111–1125.
Fershtman, C. and A. Pakes (2012, November). Dynamic games with asymmetric infor-
mation: A framework for empirical work. Quarterly Journal of Economics 127 (4),
1611–1661.
Gali, J. and M. Gertler (1999). Inflation dynamics: A structural econometric analysis.
Journal of Monetary Economics 44, 195–222.
Jaffe, S. and G. Weyl (2013). The first order approach to merger analysis. American
Economics Journal: Microeconomics 5 (4), 188–218.
Judd, K. L. (1998). Numerical Methods in Economics. London, England: The MIT
Press.
Judd, K. L. and M. H. Riordan (1994). Price and quality in a new product monopoly.
The Review of Economic Studies 61 (4), 773–789.
Kimball, M. S. (1995). The quantitative analytics of the basic neomonetarist model.
Journal of Money, Credit and Banking 27 (4), 1241–77.
Klein, B. and K. B. Leffler (1981). The role of market forces in assuring contractual
performance. Journal of Political Economy 89 (4), 615–641.
Klenow, P. J. and J. L. Willis (2006). Real rigidities and nominal price changes.
Research working paper rwp 06-03, Federal Reserve Bank of Kansas City.
29
L’Huillier, J.-P. (2013). Consumers’ imperfect information and price rigidities. Work-
ing paper.
Lucas, R. E. (1972). Expectations and the neutrality of money. Journal of Economic
Theory 4, 103–124.
Mackowiak, B. and M. Wiederholt (2009). Optimal sticky prices under rational inat-
tention. American Economic Review 99 (3), 769–803.
Mankiw, G. and R. Reis (2002). Sticky information versus sticky prices: a proposal to
replace the new keynesian phillips curve. Quarterly Journal of Economics 117 (4),
1295–1328.
Nakamura, E. and D. Zerom (2010, July). Accounting for incomplete pass-through.
Review of Economic Studies 77 (3), 1192–1230.
Null, B. (2008). The nested dirichlet distribution: Properties and applications. Work-
ing paper.
Orphanides, A. and J. Williams (2004). Imperfect Knowledge, Inflation Expecta-
tions, and Monetary Policy. ”National Bureau of Economic Research, Inc”. in
The Inflation-Targeting Debate.
Peltzman, S. (2000, June). Prices rise faster than they fall. The Journal of Political
Economy 108 (3), 466–502.
Reis, R. (2006). Inattentive producers. Review of Economic Studies 73, 793821.
Rotemberg, J. J. (2005). Customer anger at price increases, changes in the frequency
of price adjustment and monetary policy. Journal of Monetary Economics 52 (4),
829–852.
Rust, J. (1996). Numerical dynamic programming in economics. In H. Amman,
D. Kendrick, and J. Rust (Eds.), Handbook of Computational Economics. Else-
vier, North Holland.
Shapiro, C. (1983). Premiums for high quality products as returns to reputations. The
Quarterly Journal of Economics 98 (4), 659–680.
Weiser, A. and S. Zarantonello (1988). A note on piecewise linear and multilinear table
interpolation in many dimensions. Mathematics of Computation 50, 189–196.
30
Weyl, G. and M. Fabinger (2013). Pass-through as an economic tool: Principles of
incidence under imperfect competition. Journal of Political Economy 121 (3), 528–
583.
Wieland, V. (2000, August). Monetary policy, parameter uncertainty and optimal
learning. Journal of Monetary Economics 46 (1), 199–228.
Wolinsky, A. (1983). Prices as signals of product quality prices as signals of product
quality. The Review of Economic Studies 50 (4), 647–658.
31
Appendix
8 Description of the Multilinear Interpolation Algo-
rithm
We apply the algorithm in Weiser and Zarantonello (1988) to interpolate the value
function over the two dimensional continuous part of the state space. The algorithm
proceeds as follows. As we have described earlier, we split the two dimensional unit
square into an Ng by Ng regular grid, where Ng = 21. First, given a vector (α̃H , α̃L),
we figure out which grid points contain the vector: we find the integer i such that
(i − 1)/(Ng − 1) ≤ α̃H ≤ i/(Ng − 1), and the integer j such that (j − 1)/(Ng − 1) ≤
α̃H ≤ j/(Ng − 1). We then scale the square up to a [0, 1] by [0, 1] square, transforming
(α̃H , α̃L) to
(x1, x2) =
(
(Ng − 1)
(
α̃H −(i− 1)
Ng − 1
)
, (Ng − 1)
(
α̃L −(j − 1)
Ng − 1
))
.
We also relabel the save value functions at the vertices of the square as Vn(l, m), where
(l, m) ∈ {0, 1} × {0, 1}. We then figure out which simplex of the unit square contains
(x1, x2). To do this, we find a permutation of (1, 2), (p(1), p(2)) such that xp(1) ≤ xp(2).
The interpolated value function can then be constructed as a linear combination of
Vn(l, m) at the vertices using the following algorithm:
1. Start with s0 = (1, 1)
2. Let V̂ = Vn(s0)
3. Let i = 1
4. Let si = si−1−ep(i), where ep(i) is a 2-vector with 1 in position ep(i) and 0 everywhere
else.
5. Let V̂ = V̂ + (1− xp(i))(Vn(si)− Vn(si−1))
6. Increment i by 1
32
7. If i ≤ 2, go to step 4. Otherwise, return the interpolated value function V̂ .
Note that this algorithm can easily be extended to continuous state spaces with N > 2
dimensions. It can also be extended to include nonregular grids. For a nonregular grid,
the only change is that we find the grid points containing (α̃H , α̃L), and scale up the grid
points to the unit hypercube.
9 The Effect of Changes in β on the Optimal Price
(Myopic Firm)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
−0.
20.
00.
20.
40.
60.
81.
0
Optimal Pricing for β= 0.5 and β= 1, Myopic Firm
Q
P/M
R/M
C
Q/MR for β= 0.5Q/MR for β= 1MC
0.0 0.1 0.2 0.3 0.4 0.5 0.6
−0.
50.
00.
51.
0
Optimal Pricing for β= 3 and β= 10, Myopic Firm
Q
P/M
R/M
C
Q/MR for β= 3Q/MR for β= 10MC
Figure 3: Optimal Myopic Price for Different Values of β
33
10 Steady-State Simulation
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
Time
Pric
e
0 100 200 300 400 500
02
46
810
Time
Qua
lity
0 100 200 300 400 500
05
1015
Time
β
0 100 200 300 400 500
0.00
0.05
0.10
0.15
0.20
0.25
Time
σ2
Figure 4: Simulation Results, γ = ∞
34
0 100 200 300 400 500
0.0
0.2
0.4
0.6
0.8
1.0
Time
Pric
e
0 100 200 300 400 500
02
46
810
Time
Qua
lity
0 100 200 300 400 500
05
1015
Time
β
0 100 200 300 400 500
0.00
0.05
0.10
0.15
0.20
0.25
Time
σ2
Figure 5: Simulation Results, γ = 0
35
11 Robustness: Discrete Price Model
In this section we present a robustness check against the linear functional form of our
consumer learning process. For this section we assume that quality, xt, can take on one of
two values {xL, xH}, where xH > xL and that the firm can only choose a discrete number
of prices: pt ∈ {p1, ..., pK}. Consumers correctly believe that the joint distribution of
pt and xt are discrete, and update those beliefs using a multinomial-Dirichlet Bayesian
updating process.
11.1 Demand
We assume there exists a continuous measure of consumers indexed by i, each with
taste for quality vi ∼ U [0, v] and endowed with wealth wi ∼ U [0, w]. Under an additively
linear direct utility function, it follows that consumer i’s indirect utility upon purchase
of a product with vertical quality xt and price pt is:
Ui = vixt − pt, (19)
The more general case consists of the scenario where the consumer does not know the
true quality of the product, xt, and thus must infer the quality based on her information
set (that is, her beliefs). We assume that the consumer believes the relationship between
price and quality takes the form:
Prob(xt = xj , pt = pk) = µjk, (20)
where j, k ∈ {H,L}. Therefore, upon observing the good’s price, pt, the consumer forms
an expectation of its quality based on the conditional probability:
E[xt|pt = pk] =
(µHk
µHk + µLk
)
xH +
(µLk
µHk + µLk
)
xL. (21)
where µHk + µLk is the probability of observing price pk. Consumers’ beliefs about the
unknown parameters, µjk, are modeled as Dirichlet:
f(µjk) = Dir(αHH , αHL, αLH , αLL), (22)
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where αjk > 0. Note that under this assumption,
E [µjk] =αjk
∑
j,k∈{H,L} αjk
and
E
[µjk
µHk + µLk
]
=αjk
∑
j∈{H,L} αjk
.19
If consumers hypothetically live for an infinite amount of time, they will use all
past information of prices and quality over the course of the good’s life to infer α =
(αHH , αHL, αLH , αLL). The information in period t can be summarized in the vector
αt = (aHH,t, aHL,t, aLH,t, aLL,t), where consumers believe that the distribution of α at
period t is Dir(αt). By the end of period t, consumers have observed both price and
quality, and will update their beliefs using a multinomial-Dirichlet Bayesian updating
process:
αjk,t+1 =
{
αjk,t + 1 if xt = xj and pt = pk
αjk,t otherwise(23)
By updating their beliefs each period, consumers learn the relationship between price and
quality. One problematic feature of assuming an infinitely lived consumer is that, in the
limit, her beliefs about the relationship between price and quality become fixed. That
is, α will eventually be learned with zero standard error. In this case, any alteration of
the stochastic process of price and quality will not cause beliefs to update.20
To allow for beliefs to continue updating indefinitely, we propose that consumers
recall the previous period’s information with error. In our multinomial-Dirichlet Bayesian
updating problem, demand for the product will be a function of consumer beliefs about
19Because the µjk’s are Dirichlet, the random variablesµjk
µHk+µLkalso follow a Dirichlet distribution
(see Null (2008) and citations therein).20We have solved and simulated a version of our model with infinitely-lived consumers, and we find
that when consumers become sure enough of their beliefs, the firm will charge a high price but lower
product quality, and beliefs will update so slowly that the firm can maintain a low quality. In the limit,
as consumers become completely sure of their beliefs, the firm can keep quality low and price high and
consumers will continue to believe that ex-ante quality will be high.
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quality conditional on price (see Equation (21)). We therefore assume that the consumers
keep track of the conditional probabilities of observing xH given pt:
Prob(xt = xH |pt = pk,αt) =αHk,t
αHk,t + αLk,t
= α̃k,t.
The memory error, which we denote νt, subsequently affects this conditional probability
by introducing noise into the consumer’s belief about quality upon observing price. We
assume that there are two error terms, νHt and νLt, each corresponding to conditioning
on pt = pH or pt = pL, respectively. We assume that the νkt’s are i.i.d. across time and
k, and that they follow a discrete distribution,
νkt =
{
ν with probability πν
ν with probability 1− πν
(24)
We further assume that consumers have limited memory in the sense that the consumer’s
information set only includes the last Nk observations, where αHk,t + αL,k,t = Nk. In
this sense, Nk determines the weight the consumer places on the observed current price
in affecting her beliefs about quality. The modified Bayesian updating process in our
overlapping generations type model is
α̃k,t+1 = l(pt, xt, α̃k,t) =
Nk
Nk+1α̃k,t +
1Nk+1
+ Nk
Nk+1νkt if xt = xH and pt = pk
Nk
Nk+1α̃k,t +
Nk
Nk+1νkt if xt = xL and pt = pk
α̃k,t + νkt otherwise.
(25)
The intuition behind this quasi-Bayesian updating formula is as follows. We can
interpret the αjk,t’s as the number of times each quality and price combination has been
observed. Consumers have a limited amount of memory, in the sense that they can only
remember the past Nk observations for each value of pk. Every period the αjk’s are
perturbed by an error that approximately preserves the average conditional probability
α̃k,t; if the firm does not charge pk in period t, then E[α̃k,t] = E[α̃k,t+1]. If the firm does
charge pk, then an observation must be added to either the high quality or low quality
bin, depending on the value of x that is chosen. The multiplication by Nk
Nk+1in this case
keeps the total number of observations for pk fixed at Nk. Note that if the firm chooses
38
to produce the high quality good, for example, consumer beliefs about the probability
of a high quality good given pk will increase on average by a factor of 1Nk+1
, which is the
same as how much the probability would go up under the standard Bayesian updating
procedure. Note that the size of Nk will determine how quickly consumer beliefs adjust.
If Nk is large, beliefs will adjust slowly in response to new information; if it is small, new
information will receive greater weight.
Thus, in the limit, there always exists uncertainty about the underlying parameter
vector α. In every period, it is only necessary for consumers to track the state variable
α̃t = (α̃Ht, α̃Lt). It follows from (19) that consumer i will purchase the product if both
E(xt|pt, α̃t) − pt > 0 and pt < wi. Hence, the demand curve for each period that the
